Simultaneous construction of dual Borgen plots. I: The case of noise‐free data

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A major problem of multivariate curve resolution methods is the underlying non-uniqueness of the pure component decompositions. This raises the question how a chemical experiment should be designed so that the solution ambiguity is as small as possible. Changes of the reaction conditions belong to the possible variations whereas for a fixed chemical reaction system, the pure component spectra appear to be unchangeable. The paper investigates and discusses the possibility to design a chemical experiment in a way that minimizes the ambiguity of the factorization. The analysis identifies regions of the spectra that are responsible for a small ambiguity. Certain sources are identified that are responsible for an increased ambiguity by means of an a posteriori analysis. This results in recommendations how to construct spectral measurements incorporating a reduced factorization ambiguity. Furthermore, lower bounds on an unavoidable base level of ambiguity are specified under the constraint of fixed reactants. The problem analysis is accompanied by investigations of several experimental data sets. KEYWORDS ambiguity, design of experiments, multivariate curve resolutionJournal of Chemometrics. 2020;34:e3159.wileyonlinelibrary.com/journal/cem

Section: Borgen Plots and Rotational Ambiguity In Afs Constructionsmentioning

confidence: 99%

T = (\begin{array}{llll} 1 & x_{1} & ⋯ & x_{s - 1} \\ 1 \\ ⋮ & normalW \\ 1 \end{array}) .

The AFS for the factor S reads with x =( x 1 ,…, x s −1 ) T and W

M_{S} = false{x \in ℝ^{s - 1} : exists W \in ℝ^{false(s - 1 false) \times false(s - 1 false)} such that rank false(T false) = s and C, S \geq 0 false} .

Its pendant for C is denoted

M_{C}

Section: Borgen Plots and Rotational Ambiguity In Afs Constructionsmentioning

confidence: 99%

“…The polyhedra are defined as follows

\begin{matrix} right {scriptF}_{S} & left = \{x \in {double-struckR}^{s - 1} : V (\begin{array}{lllll} 1 \\ x \end{array}) \geq 0\}, & right {scriptI}_{S} & left = convhull ({a_{i}, i = 1, ⋯, k}), & right \\ right {scriptF}_{C} & left = \{y \in {double-struckR}^{s - 1} : U \sum (\begin{array}{lllll} 1 \\ y \end{array}) \geq 0\}, & right {scriptI}_{C} & left = convhull ({b_{j}, j = 1, ⋯, n}) . & right \end{matrix}

a_{i} = \frac{{false(U normal∑ false)}^{T} false(2 : s, i false)}{{false(U normal∑ false)}^{T} false(1, i false)}, i = 1, ⋯, k, and b_{j} = \frac{{false(V false(j, 2 : s false) false)}^{T}}{V false(j, 1 false)}, j = 1, ⋯, n;

see also eqs (6) and (7) in Sawall et al With these polyhedra, a simple geometric construction makes it possible to construct the AFS. For instance, the spectral AFS

M_{S}

is the set of all vertices of all ( s −1)‐simplices that include

…”

Section: Borgen Plots and Rotational Ambiguity In Afs Constructionsmentioning

confidence: 99%

“…In particular, for s =3, the 2‐simplices are triangles that enclose INNPOL and are included in Borgen and Kowalski and Rajkó and István . A simultaneous construction of

M_{S}

and

M_{C}

is possible by using certain duality relations between

I_{C}

and

F_{S}

and also

I_{S}

and

F_{C}

…”

Section: Borgen Plots and Rotational Ambiguity In Afs Constructionsmentioning

confidence: 99%

“…The numbers of facets of

F_{S}

and

F_{C}

as well as the numbers of vertices of

I_{S}

and

I_{C}

are limited by the dimensions k and n of D . Due to the duality relations, the number of facets of

F_{S}

equals the number of vertices of

I_{C}

and is smaller than or equal to n . Analogically, the number of facets of

F_{C}

equals the number of vertices of

I_{S}

and is smaller than or equal to k .…”

Section: Borgen Plots and Rotational Ambiguity In Afs Constructionsmentioning

confidence: 99%

See 3 more Smart Citations

Multivariate curve resolution methods and the design of experiments

Sawall

Kubis

Schröder

et al. 2019

Self Cite

Simultaneous construction of dual Borgen plots. II: Algorithmic enhancement for applications to noisy spectral data

Sawall

Moog

Kubis

et al. 2018

Self Cite

Borgen plots are low‐dimensional representations of the set of all nonnegative factorizations of spectral data matrices. Classical Borgen plots are limited to nonnegative data and can be constructed for the spectral factor or for the concentration profiles. In the first part of this paper, a simultaneous construction of the two dual Borgen plots is presented, which intensively exploits the underlying duality principles. The second part introduces algorithmic enhancements which make the simultaneous Borgen plot construction possible for noisy experimental data matrices which can contain small negative matrix entries. The new method is tested for FT‐IR spectral data from the Rhodium catalyzed hydroformylation process. The results are compared to those by the FACPACK‐implementation of the polygon inflation method.

Trilinear self‐modeling curve resolution using Borgen‐Rajkó plot

Omidikia

Abdollahi

Kompany‐Zareh

et al. 2019

Modern analytical instruments provide measurement data arrays with full of hidden and redundant information. Multivariate curve resolution (MCR) techniques decompose data set to physic‐chemically meaningful abstract profiles. On the other hand, for such data matrices, Borgen‐Rajkó self‐modeling curve resolution (SMCR) techniques reveal all possible solutions analytically under the minimal assumption. Although Lawton‐Sylvestre (LS) and Borgen methods have been proposed for the non‐negative curve resolution of two‐component and three‐component systems, there is still a great deal of interest to include further restrictions on the Borgen‐Rajkó SMCR. As modern hyphenated analytical instruments produce multiway (eg, three‐way) arrays, multiway analysis (eg, trilinear decomposition) was received much more popularity by chemists. This contribution is decicated to the extension of the Borgen algorithm to the trilinear data sets. The Borgen method incorporating trilinearity constraint is approached in an analogy to the trilinear Lawton‐Sylvester method. The proposed analytical triadic decomposition is applied to the simulated three‐way arrays with full rank and rank overlap (a type of rank deficiency) loadings. Finally, the proposed algorithm is further exemplified with a three‐component spectrofluorimetric study of acid‐base equilibrium of the pyridoxine. Investigating feasible regions of the simulated and experimental three‐component arrays reveal interesting additional information.